One number $H \in (0, 1)$ captures how a process's variance scales with time. $H = 0.5$ is Brownian; $H > 0.5$ is persistent; $H < 0.5$ is rough. Equity log-vol comes out at $H \approx 0.1$ — the rough-vol headline empirical fact.
Standard Brownian motion has a beautiful variance-scaling property: $\mathrm{Var}(W_t) = t$. The Hurst exponent generalises this to processes where $\mathrm{Var}(X_t) \sim t^{2H}$ for some $H \in (0, 1)$. $H = 0.5$ recovers Brownian motion. $H > 0.5$ means the process is *persistent* (positive autocorrelation in increments) — trends continue. $H < 0.5$ means the process is *anti-persistent* or *rough* — increments mean-revert. The headline empirical finding of the rough-volatility programme (Gatheral-Jaisson-Rosenbaum 2014) is that log-realised-volatility on equity-index returns has $H \approx 0.1$, dramatically below the $H = 0.5$ assumption built into Heston and every other classical stochastic-vol model.
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Worked example
A fractional Brownian motion $B^H_t$ has variance $\mathrm{Var}(B^H_t) = t^{2H}$ for Hurst exponent $H$. Suppose $\mathrm{Var}(X_1) = 1$ at time $t = 1$. (a) What is $\mathrm{Var}(X_{10})$ if $H = 0.5$? (b) What if $H = 0.1$ (the rough-vol regime)? (c) What if $H = 0.9$ (highly persistent)?
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Reflection
The roughness regime $H \approx 0.1$ implies that vol-of-vol on short horizons is enormous compared to what Heston predicts (which has effective $H = 0.5$). What concretely changes about short-dated option pricing in a rough-vol model vs a classical stochastic-vol model?